Had a deep slice on the tip of the thumb 3 days ago when cutting food. the wound is already closed up without stitches, no bleeding, no pus, no pain unless pressure applied. however, the edges of the sliced bit are purple in colour. is that normal?

Healing from injury is a process, and 'normal' is not the word for it; give enough time for healing. The skin changes may be part of healing process and you have indicated by the fact that there is not much pain that infection is unlikely. Precaution to avoid introducing bacteria by cleaning and covering when handling dirty chores should allow faster healing .